ItemTwo sides of the megalopolis: Educating for sustainable citizenship(2012) Pope, Alexander; Patterson, TimothyDespite widespread focus on literacy and math at the expense of other subjects, citizenship and environmental education have an important role in American public education. Citizenship and environmental education are broadly tasked with helping students develop the knowledge, skills, and attitudes needed to shepherd the body politic and natural world, respectively, into the future. For educators and administrators concerned with instructional efficiency, educational farm visits offer one means of pairing these two approaches into a unified learning experience. This paper presents findings from a qualitative case study analysis of two such programs, incorporating interviews with and observations of visiting students, teachers, and parents. The authors argue that sustainable citizenship—a typically European conception of citizenship that stresses the natural as well as the national world—is an important outcome of these types of educational experiences. ItemUsing the five practices model to promote statistical discourse(2015) Groth, Randall E.Statistical tasks that can be solved in a variety of ways provide rich sites for classroom discourse. Orchestrating such discourse requires careful planning and execution. Stein, Engle, Smith, and Hughes (2008) suggested five practices to help teachers do so. The five practices can be used to structure conversations so that coherent classroom narratives about solutions to tasks may be formed. In this manuscript, two classroom examples that illustrate the five practices are offered. It is argued that employing the five practices can lead to higher quality classroom discussion than some commonly used arrangements. ItemHigh school students’ levels of thinking in regard to statistical study design(2003) Groth, Randall E.The study describes levels of thinking in regard to the design of statistical studies. Clinical interviews were conducted with 15 students. Each student was enrolled in high school or was a recent graduate. The students interviewed represented a range of mathematical backgrounds. During the clinical interview sessions, students were asked how they would go about designing studies to answer several different quantifiable questions. Several levels of sophistication were identified in their responses. The levels of sophistication in response are discussed in terms of the Biggs and Collis (1982, 1991) cognitive model. ItemAn investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value(2005) Groth, Randall E.The study describes students’ patterns thinking for statistical problems set in two different contexts. Fifteen students representing a wide range of experiences with high school mathematics participated in problem-solving clinical interview sessions. At one point during the interviews, each solved a problem that involved determining the typical value within a set of incomes. At another point, they solved a problem set in a signal-versus-noise context (Konold & Pollatsek, 2002). Several patterns of thinking emerged in the responses to each task. In responding to the two tasks, some students attempted to incorporate formal measures, while others used informal estimating strategies. The different types of thinking employed in using formal measures and informal estimates are described. The types of thinking exhibited in the signal-versus-noise context are then compared against those in the typical value context. Students displayed varying amounts of attention to both data and context in formulating responses to both problems. Suggestions for teachers in regard to helping students attend to both data and context when analyzing statistical data are given. ItemAn exploration of students' statistical thinking(2006) Groth, Randall E.The statistical thinking exhibited by 14-19 year old students during clinical interview sessions is described. The students’ thinking in regard to fundamental statistics concepts is reported in order to help inform instructional practice. ItemPreservice elementary teachers' conceptual and procedural knowledge of mean, median, and mode(2006) Groth, Randall E.; Bergner, Jennifer A.The paper describes aspects of the statistical content knowledge of 46 preservice elementary school teachers. The preservice teachers responded to a written item designed to assess their knowledge of mean, median, and mode. The data produced in response to the written item were examined in light of the Structure of the Observed Learning Outcome (SOLO) Taxonomy (Biggs & Collis, 1982, 1991) and Ma’s (1999) conception of Profound Understanding of Fundamental Mathematics (PUFM). Four levels of thinking in regard to comparing and contrasting mean, median, and mode are described. Several different categories of written definitions for each measure of central tendency are also described. Connections to previous statistical thinking literature are discussed, implications for teacher education are given, and directions for further research are suggested. ItemTeachers' perspectives on mathematics education research reports(2007) Groth, Randall E.; Bergner, Jennifer A.Practicing teachers’ perspectives on a set of mathematics education research reports are described. Data were gathered through e-mail messages, group discussions, and questionnaires. Teachers identified positive influences of research on practice aligned with some of the strands of proficient mathematics teaching identified by Kilpatrick, Swafford, and Findell (2001). Teachers also gave negative critiques of the ability of research to influence practice aligned with Kennedy’s (1997) discussion of historical factors underlying the gap between educational research and practice. The variety of perspectives documented provides some empirical ground for informing actions in the areas of teacher education and research. ItemUnderstanding teachers' resistance to the curricular inclusion of alternative algorithms(2007) Groth, Randall E.This study focuses on a group of practitioners from a school district that adopted reform-oriented curriculum materials but later rejected them, partially due to the inclusion of alternative algorithms in the materials. Metaphors implicit in a conversation among the group were analysed to illuminate their perspectives on instructional issues surrounding alternative algorithms. Several possible sources of resistance to folding alternative algorithms into instruction were found, including the ideas that: successful learning does not involve struggling with mathematics, the teacher’s role in the classroom is primarily to present information, and that mathematics learning progresses according to a fixed sequence of levels. ItemCharacteristics of teachers' conversations about teaching mean, median, and mode(2009) Groth, Randall E.The study analyzed a conversation among a group of teachers responsible for teaching the concepts of mean, median, and mode. After reading an article describing some specific student difficulties in learning the concepts, teachers were asked to discuss how the teaching of the concepts could be improved. Several claims pertinent to improving teaching practice were offered. Claims focused on the appropriate age at which to introduce statistical concepts, the influence of the state-prescribed curriculum on teaching practice, content-specific teaching strategies, and content-independent teaching strategies. Teachers’ claims were discussed in terms of points of departure and agreement with existing empirical research. ItemAn exploration of two online approaches to mathematics teacher education(2009) Groth, Randall E.; Burgess, Claudia R.The purpose of the present study was to examine the nature of the discourse generated by two different online approaches to mathematics teacher professional development. Thirty mathematics teachers participated in online activities involving analysis and discourse about artifacts of teaching practice. Half were randomly assigned to a group that analyzed and discussed students work samples, and the other half to a group focused upon descriptions of classroom teaching episodes. Teachers formed threads of conversation on asynchronous discussion boards as they considered the different aspects of each artifact. Within the threads, different orientations toward the reform agenda in mathematics education were shown. Some messages were strongly rooted in the reform paradigm, others in the traditional paradigm, and many others contained elements characteristic of each paradigm. The distribution and characteristics of the messages reflected teachers’ frequent attempts to try to reconcile the largely incompatible paradigms undergirding reform-oriented and traditional approaches to mathematics instruction. This paper describes the nature of the discourses occurring within each group of teachers with the aim of providing empirical grounds to inform the actions of teacher educators and researchers designing online learning environments that attempt to bring teachers more fully into the reform-oriented discourse about teaching mathematics. ItemCharacterizing key developmental understandings and pedagogically powerful ideas within a statistical knowledge for teaching framework(2013) Groth, Randall E.A hypothetical framework to characterize statistical knowledge for teaching (SKT) is described. Empirical grounding for the framework is provided by artifacts from an undergraduate course for prospective teachers that concentrated on the development of SKT. The theoretical notion of “key developmental understanding” (KDU) is used to identify landmarks in the development of SKT subject matter knowledge. Sample KDUs are given for the subject matter knowledge categories of common content knowledge, specialized content knowledge, and horizon knowledge. The theoretical notion of “pedagogically powerful idea” is used to describe how KDUs must be transformed to become useful in teaching. Examples of pedagogically powerful ideas for the pedagogical content knowledge categories of knowledge of content and teaching and curriculum knowledge are provided. Knowledge of content and students is hypothesized as a basis for the development of pedagogically powerful ideas. ItemMapping the structure of knowledge for teaching nominal categorical data analysis(2013) Groth, Randall E.; Bergner, Jennifer A.This report describes a model for mapping cognitive structures related to content knowledge for teaching. The model consists of knowledge elements pertinent to teaching a content domain, the nature of the connections among them, and a means for representing the elements and connections visually. The model is illustrated through empirical data generated as prospective teachers were in the process of developing knowledge for teaching nominal categorical data analysis. During a course focused on the development of statistical knowledge for teaching, the prospective teachers analyzed statistical problems, descriptions of children’s statistical thinking, and related classroom scenarios. Their analyses suggested various types of knowledge structures in development. In some cases, they constructed all knowledge elements targeted in the course. In many cases, however, their knowledge structures had missing, incompatible, and/or disconnected elements preventing them from carrying out recommendations for teaching elementary nominal categorical data analysis in an optimal manner. The report contributes to teacher education by drawing attention to prospective teachers’ learning needs, and it contributes to research on teachers’ cognition by providing a method for modeling their cognitive structures. ItemA day in the life of a statistical knowledge for teaching course(2013) Groth, Randall E.Teachers’ statistical learning needs differ from those of individuals in other professions. Along with learning statistical content, they must develop the ability to teach statistics to others. This paper illustrates how both needs can be addressed simultaneously in an undergraduate course grounded in a statistical knowledge for teaching framework. A typical day in the course is presented to demonstrate application of the framework. It is also suggested that practicing teachers can benefit from using the framework to set their own professional development goals.